The advanced part of A treatise on the dynamics of a system of rigid bodies : being part II. of a treatise on the whole subject, with numerous examples

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THE ADVANCED PART

OF A TREATISE ON THE

DYNAMICS OF A SYSTEM OF
RIGID BODIES.

BEING PART II. OF A TREATISE ON THE WHOLE
SUBJECT.

With numerous Examples.

BY

EDWARD JOHN ROUTH, DSc. LL.D. F.RS., &.

FELLOW OF THE UNIVERSITY OF LONDON;
HONORARY FELLOW OF ST PETER’S COLLEGE, CAMBRIDGE,

FOURTH EDITION, REVISED AND ENLARGED.

London:
MACMILLAN AND CO.
1884

[All Rights reserved.]

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MATH /
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PREFACE.

_ Tuts volume is intended to be a continuation of that already
“published as Part I. in 1882. The time occupied in its pre-
‘paration has been longer than I had anticipated. This is partly
due to the want of sufficient leisure, and partly also because as
I proceeded with the work new questions to which no sufficient
answers had yet been given seemed continually to arise. The
‘pleasure and labour of attempting to answer these, however im-
perfectly, has delayed the book.

_ Although a large portion of this volume has already appeared
in the latter half of the third edition, yet much of this has been
recast and new illustrations and explanations have been given
wherever they appeared to be necessary. Besides this much
new matter has been added. Exactly also as in the last edition
those parts to which the student should first turn his attention
_are printed in a larger type than the rest.

___ Following the same plan as in Vol. I., the several Chapters
have been made as independent as possible. The object in view
was that the reader should select his own order of study. His-
_ torical notices and references have been given throughout the
book. But it has not been thought necessary to refer to the
author's own additions to the subject, except when they have
been first published elsewhere.

_ In this volume much use has been made of the new symbol
for a fraction lately introduced by Prof. Stokes. The symbol
R. D. IL. b

02322

Vi PREFACE.

a/b for ; is very convenient as it enables the algebraical formule

to be written on a line with the type. If some such abbreviation
as this is not used two whole lines are required to write the
simplest fraction. When the numerator or denominator of the
fraction so written contains several factors, the rule adopted has
been that all that follows the slant line up to the next plus or
minus sign is to be regarded as the denominator. In the same
way all that precedes the slant line up to the next plus or minus
sign is to be taken as the numerator. When more complicated
factors have to be written, page are used to indicate the

numerator and denominator. Thus ~ = ite : Fs would be written

abjed + (e+ f(g - x

Numerous examples have been given throughout the book.
Some of these are intended to be merely simple exercises, but
many are important as illustrating and completing the theories
given in the text. Sometimes when the principles of a theory
had been explained numerous applications seemed to arise. In-
stead of loading the text with these it appeared preferable to
put them into the form of examples and to give such hints as
would make their solution easy. Everywhere the results have
been given, and care has been taken to secure their accuracy ;
but amongst so many problems, it cannot be expected that no
errors have escaped detection.

EDWARD J. ROUTH.

PETERHOUSE,
August, 1884.

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CONTENTS.

CHAPTER I.
MOVING AXES AND RELATIVE MOTION.

Moving axes > . : ‘
On Relative Motion and Clairaut’ 8 Thame 5 ; fA
Motion relative to the earth z f . A ‘ ;

CHAPTER II.
OSCILLATIONS ABOUT EQUILIBRIUM.

Lagrange’s method with indeterminate multipliers . F
Theorems on Lagrange’s determinant . :

Energy of an oscillating system . 2 ‘ :
Effect of changes in the system . ; ‘ ‘ : :
Composition and analysis of oscillations . - r ;

CHAPTER III.

OSCILLATIONS ABOUT A STATE OF MOTION.

The energy test of stability . : ‘ ‘ :
Examples of oscillations about steady motion. The Governor
and Laplace’s Three Particles, &, . ‘ . eer
Theory of oscillations about steady motion. é . °
The representative nga 4 j F ; F

PAGES
1—13
13—18
18—30

31—36
36—41
41—43
43—45
45—51

52—57

57—62
62—69
69—73

viii

ARTS,

141—143,
144—156,
157—175,

176—179.
180—183,
.184—191,
192—198,

199,

200—214,
215—239,

240—253,
254—255,

NATURE OF THE MOTION GIVEN BY LINEAR EQUATIONS AND

256—285,

286—307.

CONTENTS,

CHAPTER IV.

MOTION OF A BODY UNDER THE ACTION OF NO FORCES.

Solution of Euler’s Equations . ° .

Poinsot’s and MacCullagh’s constructions for the fection :

On the cones described by the invariable and instantaneous
axes; treated by Spherical Trigonometry

Motion of the Principal Axes.

Two principal moments equal

Motion when G?=BT.

Correlated and contrarelated bodies

The Sphero-conic or Spherical Ellipse

Examples . ; 3 5 :

CHAPTER V.

MOTION OF A BODY UNDER ANY FORCES.

Motion of a Top . ‘i . : .
Motion of a sphere on various rita or rough =
Billiard balls . ‘ . . ° °

Motion of a solid body on & aout or rough plane
Motion ofarod . ; :
Examples

CHAPTER VI,

CONDITIONS OF STABILITY,

Solution of differential equations with single, double, triple,
&c,, types. The conditions that all powers of the time
are bos ‘ .

The conditions of stability, (1) for a viquadrtio and (2) for
an equation of the nth degree . : , ‘

CHAPTER VIL.

FREE AND FORCED OSCILLATIONS.

808—322, On Free Oscillations, with two propositions to determine

323—354,

855—364,

their nature . é
On Forced Oscillations ; hoe inaghited or diminished, with

Herschel’s theorem on their period, &¢. , A
Second approximations . . . . . ,

PAGES
74—TT
77—86

86—95
95—98
98—99
99—104
104—108
108—109
109—110

111—122
123—138
138—150

150—151
151—153

THE

154—166

166—176

365—375.
-376—397.
[ 393—400,

- 401—419,

420421,
422441.

442462,
- 463—476.

— 481—488,
—-489—504.
—~605—514.

‘515—521.
522525,
526—535,

-536—547.

548558,

CONTENTS.

CHAPTER VIII.

INITIAL CONDITIONS.

i’ DETERMINATION OF THE CONSTANTS OF INTEGRATION IN TERMS OF THE
: ARTS

Method of Isolation . ‘ é - & er eee ;
Method of Multipliers .
Fourier’s Rule , 3

CHAPTER IX.

The Solution of Problems illustrating the two kinds of
motion . 5 \ 2 ; : F a :

Network of Particles . p 4 ;

Theory of equations of differences, with Sturm’ 8 ‘theorems.

CHAPTER X.

APPLICATIONS OF THE CALCULUS OF VARIATIONS,
Principles of Least Action and Varying Action . ;
Hamilton’s solution of the general equations of motion with

Jacobi’s complete integral : ‘ re ‘ . ;
Variation of the elements . - : ; ; 3 P

CHAPTER XI.
PRECESSION AND NUTATION,

On the Potential . ‘ ‘ ‘ : ‘ “ .
Motion of the Earth about its centre of pecit s . 3 .
Motion of the Moon about its centre of gravity . ° ‘

CHAPTER XII.
MOTION OF A STRING OR CHAIN.

Equations of Motion . ‘ ‘ : : . .

On steady motion 5 > ‘ ; : A . F
On initial motions é - x ‘ ; ; .
Small oscillations of a loose sihiates ‘i - é Z ‘

Small oscillations of a tight string . A e r

CHAPTER XIII,
MOTION OF A MEMBRANE.

Transverse oscillations of a homogeneous membrane
Motion of a heterogeneous membrane. fs

.
-

ix

PAGES
202—210
210—223
223—225

APPLICATIONS OF THE CALCULUS OF FINITE DIFFERENCES.

226—235
235—236

236—243

244—254

254—262
262—264

265—272
272—286
286—294

295—299
299—302
302—307
307—316
316—324

325—329
330—335